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HMMT 二月 2001 · 团队赛 · 第 11 题

HMMT February 2001 — Team Round — Problem 11

专题
Discrete Math / 离散数学
难度
L3
来源
HMMT

题目详情

  1. Define a ? = ( a − 1) / ( a + 1) for a 6 = − 1. Determine all real values N for which ( N ?)? = tan 15.
解析
  1. Define a ? = ( a − 1) / ( a + 1) for a 6 = − 1. Determine all real values N for which ( N ?)? = tan 15. Solution: Let x = N ?. Then ( x − 1) cos 15 = ( x + 1) sin 15. Squaring and rearranging √ √ 2 2 3 2 terms, and using the fact that cos 15 − sin 15 = cos 30 = , we have 3 x − 4 3 x + 3 = 0. 2 √ √ 3 Solving, we find that x = 3 or . However, we may reject the second root because it 3 √ √ √ 1+ x 1+ 3 √ yields a negative value for ( N ?)?. Therefore x = 3 and N = = = − 2 − 3 . 1 − x 1 − 3